Binary and Decimal Conversion in Digital Fundamentals
In the field of digital electronics and computer science, binary and decimal conversions are essential concepts. Binary (base-2) is the language of computers, while decimal (base-10) is the numeric system commonly used by humans. Understanding the conversions between these systems is crucial for computer programming, data storage, and digital circuits.
Method to Convert Binary to Decimal
- Step 1: Write down the binary number.
- Step 2: Starting from the rightmost digit, multiply each binary digit by 2 raised to the power of its position (starting from 0).
- Step 3: Sum all the products from Step 2 to get the decimal equivalent.
- Example: Binary 1011 = (1×2^3) + (0×2^2) + (1×2^1) + (1×2^0) = 11 in decimal.
Method to Convert Decimal to Binary
- Step 1: Divide the decimal number by 2 and note the remainder.
- Step 2: Write down the quotient and repeat the division by 2, noting each remainder, until the quotient is zero.
- Step 3: Write down all remainders in reverse order to get the binary equivalent.
- Example: Decimal 11 divided by 2 gives binary 1011.
Advantages and Disadvantages of Binary and Decimal Systems
| Advantages | Disadvantages |
|---|---|
| Binary simplifies digital circuit design. | Binary numbers are difficult for humans to interpret. |
| Decimal is easy for humans to read and understand. | Decimal requires more complex circuits for computers. |
| Binary systems are more efficient for computers. | Decimal calculations can slow down computer processes. |
| Binary code is essential for modern computing systems. | Decimal needs conversion to binary for computer processing. |
| Binary allows easier representation of data in logic gates. | Decimal-based devices require complex circuitry. |
| Binary values are more resilient to error in digital systems. | Decimal operations are challenging in digital devices. |
Applications of Binary and Decimal Conversion
- Binary is foundational in programming and digital systems as it is used to store, process, and communicate data within computers.
- Each bit in binary represents a state of either ON (1) or OFF (0), simplifying the physical and logical design of circuits.
- Decimal conversions are used when data must be interpreted in a human-readable format, such as displaying data on user interfaces, calculators, or performing arithmetic operations in accounting software.
- Conversions between these systems are also critical in areas like network addressing (e.g., IP addresses), digital image processing, and data encryption.
Question 1: Convert decimal number 10 to binary.
Solution: The decimal number 10 can be converted to binary by dividing by 2 and writing down the remainders.
- 10 ÷ 2 = 5, remainder 0
- 5 ÷ 2 = 2, remainder 1
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
Now, read the remainders from bottom to top: 1010 is the binary equivalent of decimal 10.
Question 2: Convert binary number 1011 to decimal.
Solution: The binary number 1011 can be converted to decimal by multiplying each bit by 2 raised to the power of its position (starting from 0 on the right).
- (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (1 × 2^0)
- 8 + 0 + 2 + 1 = 11
The decimal equivalent of binary 1011 is 11.
Question 3: Convert decimal number 25 to binary.
Solution: Divide 25 by 2, writing down the remainders.
- 25 ÷ 2 = 12, remainder 1
- 12 ÷ 2 = 6, remainder 0
- 6 ÷ 2 = 3, remainder 0
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 11001 is the binary equivalent of decimal 25.
Question 4: Convert binary number 1100 to decimal.
Solution: Multiply each bit by 2 raised to the power of its position.
- (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (0 × 2^0)
- 8 + 4 + 0 + 0 = 12
The decimal equivalent of binary 1100 is 12.
Question 5: Convert decimal number 47 to binary.
Solution: Divide 47 by 2, writing down the remainders.
- 47 ÷ 2 = 23, remainder 1
- 23 ÷ 2 = 11, remainder 1
- 11 ÷ 2 = 5, remainder 1
- 5 ÷ 2 = 2, remainder 1
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 101111 is the binary equivalent of decimal 47.
Question 6: Convert binary number 101010 to decimal.
Solution: Multiply each bit by 2 raised to the power of its position.
- (1 × 2^5) + (0 × 2^4) + (1 × 2^3) + (0 × 2^2) + (1 × 2^1) + (0 × 2^0)
- 32 + 0 + 8 + 0 + 2 + 0 = 42
The decimal equivalent of binary 101010 is 42.
Question 7: Convert decimal number 63 to binary.
Solution: Divide 63 by 2, writing down the remainders.
- 63 ÷ 2 = 31, remainder 1
- 31 ÷ 2 = 15, remainder 1
- 15 ÷ 2 = 7, remainder 1
- 7 ÷ 2 = 3, remainder 1
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 111111 is the binary equivalent of decimal 63.
Question 8: Convert binary number 100101 to decimal.
Solution: Multiply each bit by 2 raised to the power of its position.
- (1 × 2^5) + (0 × 2^4) + (0 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0)
- 32 + 0 + 0 + 4 + 0 + 1 = 37
The decimal equivalent of binary 100101 is 37.
Question 9: Convert decimal number 15 to binary.
Solution: Divide 15 by 2, writing down the remainders.
- 15 ÷ 2 = 7, remainder 1
- 7 ÷ 2 = 3, remainder 1
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
Reading the remainders from bottom to top: 1111 is the binary equivalent of decimal 15.
Question 10: Convert binary number 1110 to decimal.
Solution: Multiply each bit by 2 raised to the power of its position.
- (1 × 2^3) + (1 × 2^2) + (1 × 2^1) + (0 × 2^0)
- 8 + 4 + 2 + 0 = 14
The decimal equivalent of binary 1110 is 14.